Objective. Evaluate the effectiveness of cognitive-behavior therapy for chronic fatigue syndrome.
Participant pool. 142 patients who were recruited from referrals by primary care physicians and consultants to a hospital clinic specializing in chronic fatigue syndrome.
Actual participants. Only 60 of the 142 referred patients entered the study. Some were excluded because they didn't meet the diagnostic criteria, some had other health issues, and some refused to be a part of the study.
Patients randomly assigned to treatment and control groups, 30 patients in each group:
Treatment: Cognitive behavior therapy \(-\) collaborative, educative, and with a behavioral emphasis. Patients were shown on how activity could be increased steadily and safely without exacerbating symptoms.
Control: Relaxation \(-\) No advice was given about how activity could be increased. Instead progressive muscle relaxation, visualization, and rapid relaxation skills were taught.
The table below shows the distribution of patients with good outcomes at 6-month follow-up. Note that 7 patients dropped out of the study: 3 from the treatment and 4 from the control group.
Proportion with good outcomes| Yes | No | Total | |
|---|---|---|---|
| Groups | |||
| Treatment | 19 | 8 | 27 |
| Control | 5 | 21 | 26 |
| Total | |||
| 24 | 29 | 53 | |
Treatment Group: 19/27 = 0.70 = 70%
Control Group: 5/26 = 0.19 = 19%
Do the data show a "real" difference between the groups?
Suppose you flip a coin 100 times. While the chance a coin lands heads in any given coin flip is 50%, we probably won't observe exactly 50 heads. This type of fluctuation is part of almost any type of data generating process.
The observed difference between the two groups (70 - 19 = 51%) may be real, or may be due to natural variation.
Since the difference is quite large, it is more believable that the difference is real.
We use statistical tools to determine if the difference is so large that we should reject the notion that it was due to chance.
Are the results of this study generalizable to all patients with chronic fatigue syndrome?
No. These patients had specific characteristics and volunteered to be a part of this study, therefore they may not be representative of all patients with chronic fatigue syndrome.
While we cannot immediately generalize the results to all patients, this first study is encouraging. The method at least works for patients with some narrow set of characteristics, and that gives hope that it will work, at least to some degree, with other patients.
Data collected on students in a statistics class on a variety of variables
What type of variable is a telephone area code?
What type of variable is a telephone area code?
Does there appear to be a relationship between the hours of study per week and the GPA of a student?
Can you spot anything unusual about any of the data points?
Based on the scatterplot, which of the following statements is correct about the head and skull length of possums?
Based on the scatterplot, which of the following statements is correct about the head and skull length of possums?
Research Question Can people become better, more efficient runners on their own, merely by running?
Population of Interest All people
Sample Group of adult women who recently joined a running group
Population to which results can be generalized Adult women, if the data are randomly sampled
Non-response: If only a small fraction of the randomly sampled people choose to respond to a survey, the sample may no longer be representative of the population.
Voluntary response: Occurs when the sample consists of people who volunteer to respond because they have strong opinions on the issue. Such a sample will also not be representative of the population.
Convenience sample: Individuals who are easily accessible are more likely to be included in the sample.
A historical example of a biased sample yielding misleading results. In 1936, Alf Landon became the Republican presidential nominee, opposing the re-election of Franklin Delano Roosevelt, a Democrat.
Landon (GOP)
FDR (DEM)
These groups had incomes well above the national average of the day (remember, this is Great Depression era) which resulted in lists of voters far more likely to support Republicans than a truly typical voter of the time, i.e., the sample was not representative of the American population at the time.
A school district is considering whether it will no longer allow high school students to park at school after two recent accidents where students were severely injured. As a first step, they survey parents by mail, asking them whether or not the parents would object to this policy change. Of 6,000 surveys that go out, 1,200 are returned. Of these 1,200 surveys that were completed, 960 agreed with the policy change and 240 disagreed. Which of the following statements are true?
A school district is considering whether it will no longer allow high school students to park at school after two recent accidents where students were severely injured. As a first step, they survey parents by mail, asking them whether or not the parents would object to this policy change. Of 6,000 surveys that go out, 1,200 are returned. Of these 1,200 surveys that were completed, 960 agreed with the policy change and 240 disagreed. Which of the following statements are true?
Observational study: Researchers collect data in a way that does not directly interfere with how the data arise, i.e. they merely "observe", and can only establish an association between the explanatory and response variables.
Experiment: Researchers randomly assign subjects to various treatments in order to establish causal connections between the explanatory and response variables.
If you're going to walk away with one thing from this class, let it be "correlation does not imply causation".
New study sponsored by General Mills says that eating breakfast makes girls thinner
By ALEX DOMINGUEZ, Associated Press
Girls who regularly ate breakfast, particularly one that includes cereal, were slimmer than those who skipped the morning meal, according to a study that tracked nearly 2,400 girls for 10 years.
Girls who ate breakfast of any type had a lower average body mass index, a common obesity gauge, than those who said they didn't. The index was even lower for girls who said they ate cereal for breakfast, according to findings of the study conducted by the Maryland Medical Research Institute. The study received funding from the National Institutes of Health and cereal-maker General Mills.
"Not eating breakfast is the worst thing you can do, that's really the take-home message for teenage girls," said study author Bruce Barton, the Maryland institute's president and CEO.
The fiber in cereal and healthier foods that normally accompany cereal, such as milk and orange juice, may account for the lower body mass index among cereal eaters, Barton said.
The results were gleaned from a larger NIH survey of 2,379 girls in California, Ohio and Maryland who were tracked between ages 9 and 19. Results of the study appear in the September issue of the Journal of the American Dietetic Association.
Nearly one in three adolescent girls in the United States is overweight, according to the association. The problem is particularly troubling because research shows becoming overweight as a child can lead to a lifetime struggle with obesity.
As part of the survey, the girls were asked once a year what they had eaten during the previous three days. The data were adjusted to compensate for factors such as differences in physical activity among the girls and normal increases in body fat during adolescence.
"Girls who regularly ate breakfast, particularly one that includes cereal, were slimmer than those who skipped the morning meal, according to a study that tracked nearly 2,400 girls for 10 years. […] As part of the survey, the girls were asked once a year what they had eaten during the previous three days."
This is an observational study since the researchers merely observed the behavior of the girls (subjects) as opposed to imposing treatments on them.
There is an association between girls eating breakfast and being slimmer.
General Mills (cereal manufacturer)
A prospective study identifies individuals and collects information as events unfold.
Retrospective studies collect data after events have taken place.
Almost all statistical methods are based on the notion of implied randomness.
If observational data are not collected in a random framework from a population, these statistical methods – the estimates and errors associated with the estimates – are not reliable.
Most commonly used random sampling techniques are simple, stratified, and cluster sampling.
Randomly select cases from the population, where there is no implied connection between the points that are selected.
Strata are made up of similar observations. We take a simple random sample from each stratum.
Clusters are usually not made up of homogeneous observations, and we take a simple random sample from a random sample of clusters. Usually preferred for economical reasons.
A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments. Which approach would likely be the least effective?
A city council has requested a household survey be conducted in a suburban area of their city. The area is broken into many distinct and unique neighborhoods, some including large homes, some with only apartments. Which approach would likely be the least effective?
Studies where researchers assign treatments to cases are called experiments. If the assignment of treatments to cases (e.g., using a coin flip to determine which treatment a patient receives), the study is called a randomized experiment.
Randomized experiments have a series of four principles.
Researchers assign treatments to cases, and do their best to control for other differences between groups.
Example: in a drug trial, patients may be asked to take a pill daily. Some may take the pill dry (ick!), some with just a sip of water, some with coffee, and others with juice. To control for the effect of accompanying liquid, a doctor may ask all patients to drink a 12 oz glass of water with the pill.
Researchers randomize patients into treatment groups to account for variables that cannot be controlled.
Example: some patients are more susceptible to disease than others due to dietary habits. Randomizing patients into treatment/control groups helps even out these differences, possibly preventing accidental bias.
The more cases researchers observe, the more accurately they can estimate the effects of explanatory variables on response variables. In a single study, we replicate by collecting a sufficiently large sample. Additionally, scientists often replicate an entire study over again to verify earlier findings.
Researchers sometimes know (or suspect) that variables other than the treatment influence the response. Under this situation, they may first group individuals by this variable, and then randomize cases within each block. This is known as blocking.
Example: If we were researching the effect of a drug on heart attacks, we might first split patients into high-risk and low-risk blocks (based on diet, physique, genetic screening, or some other approach), and then randomly assign half of each block to the control group, and the other half to the drug (treatment) group.
We would like to design an experiment to investigate if energy gels makes you run faster:
It is suspected that energy gels might affect pro and amateur athletes differently, therefore we block for pro status:
Why is this important? Can you think of other variables to block for?
A study is designed to test the effect of light level and noise level on exam performance of students. The researcher also believes that light and noise levels might have different effects on males and females, so wants to make sure both genders are equally represented in each group. Which of the below is correct?
A study is designed to test the effect of light level and noise level on exam performance of students. The researcher also believes that light and noise levels might have different effects on males and females, so wants to make sure both genders are equally represented in each group. Which of the below is correct?}
Factors are conditions we can impose on the experimental units.
Blocking variables are characteristics that the experimental units come with, that we would like to control for.
Blocking is like stratifying, except used in experimental settings when randomly assigning, as opposed to when sampling.
Placebo: fake treatment, often used as the control group for medical studies
Placebo effect: experimental units showing improvement simply because they believe they are receiving a special treatment
Blinding: when experimental units do not know whether they are in the control or treatment group
Double-blind: when both the experimental units and the researchers who interact with the patients do not know who is in the control and who is in the treatment group
What is the main difference between observational studies and experiments?
What is the main difference between observational studies and experiments?
Scatterplots are useful for visualizing the relationship between two numerical variables.
Do life expectancy and total fertility appear to be associated or independent?
They appear to be linearly and negatively associated: as fertility increases, life expectancy decreases.
Was the relationship the same throughout the years, or did it change?
The relationship changed over the years.
Useful for visualizing one numerical variable. Darker colors represent areas where there are more observations.
How would you describe the distribution of GPAs in this data set?
Make sure to say something about the center, shape, and spread of the distribution.
The mean, also called the average (marked with a triangle in the plot), is one way to measure the center of a distribution of data.
The mean GPA is 3.59.
The sample mean, denoted as \(\bar{x}\), can be calculated as \[ \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{1}{n} \sum_{i=1}^{n} x_i \] where \(x_1, x_2, \cdots, x_n\) represent the \(n\) observed values.
The population mean is also computed the same way but is denoted as \(\mu\). It is often not possible to calculate \(\mu\) since population data are rarely available.
The sample mean is a sample statistic, and serves as a point estimate of the population mean. This estimate may not be perfect, but if the sample is good (representative of the population), it is usually a pretty good estimate.
Higher bars represent areas where there are more observations, makes it a little easier to judge the center and the shape of the distribution.
Which one(s) of these histograms are useful? Which reveal too much about the data? Which hide too much?
Does the histogram have a single prominent peak (unimodal), several prominent peaks (bimodal/multimodal), or no apparent peaks (uniform)?
Note: In order to determine modality, step back and imagine a smooth curve over the histogram – imagine that the bars are wooden blocks and you drop a limp spaghetti over them, the shape the spaghetti would take could be viewed as a smooth curve.
Is the histogram right skewed, left skewed or symmetric?
Histograms are said to be skewed to the side of the long tail.
Are there any unusual observations or potential outliers?
Which of these variables do you expect to be uniformly distributed?
Which of these variables do you expect to be uniformly distributed?
Sketch the expected distributions of the following variables:
Come up with a concise way (1-2 sentences) to teach someone how to determine the expected distribution of any variable.
How useful are centers alone for conveying the true characteristics of a distribution?
Variance is roughly the average squared deviation from the mean.
\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
\[ \begin{split} s^2 &= \frac{1}{217 - 1} \left[ (5-6.71)^2 + (9-6.71)^2 + \cdots + (7-6.71)^2 \right] \\ &= 4.11 \text{ hours}^2 \end{split} \]
Why do we use the squared deviation in the calculation of variance?
The standard deviation is the square root of the variance, and has the same units as the data.
\[ s = \sqrt{s^2} \]
The standard deviation of the amount of sleep students get per night can be calculated as: \[ s = \sqrt{4.11} = 2.03 \text{ hours} \] We can see that all of the data are within 3 standard deviations of the mean of \(\bar{x} = 6.17\).
The median is the value that splits the data in half when ordered in ascending order.
If there are an even number of observations, then the median is the average of the two values in the middle.
Since the median is the midpoint of the data, 50% of the values are below it. Hence, it is also the 50th percentile.
A percentile is the the smallest value from an ordered list of numbers which is greater than or equal to that percentage of list elements.
Example: The \(42^\text{nd}\) percentile of the numbers \(\{ 1, 2, 3, \cdots, 99, 100 \}\) is 42.
It can become quite complicated when there aren't an even multiple of 100 items!
Between Q1 and Q3 is the middle 50% of the data. The range these data span is called the interquartile range, or the IQR.
\[ \text{IQR} = \text{Q3} - \text{Q1} \]
The box in a box plot represents the middle 50% of the data, and the thick line in the box is the median.
The whiskers of a box plot can extend up to \(1.5 \times \text{IQR}\) away from the quartiles.
Example: IQR: 20 - 10 = 10
A potential outlier is defined as an observation beyond the maximum reach of the whiskers. It is an observation that appears extreme relative to the rest of the data.
Why is it important to look for outliers?
How would sample statistics such as mean, median, SD, and IQR of household income be affected if the largest value was replaced with $10 million? What if the smallest value was replaced with $10 million?
Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,
If you would like to estimate the typical household income for a student, would you be more interested in the mean or median income?
Median and IQR are more robust to skewness and outliers than mean and SD. Therefore,
If you would like to estimate the typical household income for a student, would you be more interested in the mean or median income?
Median
If the distribution is symmetric, center is often defined as the mean:
If the distribution is skewed or has extreme outliers, center is often defined as the median
Which is most likely true for the distribution of percentage of time actually spent taking notes in class versus on Facebook, Twitter, etc.?
Which is most likely true for the distribution of percentage of time actually spent taking notes in class versus on Facebook, Twitter, etc.?
If we compute, the mean = 80% and the median = 76%. So …
When data are extremely skewed, transforming them might make modeling easier. A common transformation is the log transformation.
The histograms on the left shows the distribution of number of basketball games attended by students. The histogram on the right shows the distribution of log of number of games attended.
Skewed data are easier to model with when they are transformed because outliers tend to become far less prominent after an appropriate transformation.
| # of games | 70 | 50 | 25 | \(\cdots\) |
|---|---|---|---|---|
| # of games | 4.25 | 3.91 | 3.22 | \(\cdots\) |
However, results of an analysis might be difficult to interpret because the log of a measured variable is usually meaningless.
What other variables would you expect to be extremely skewed?
Skewed data are easier to model with when they are transformed because outliers tend to become far less prominent after an appropriate transformation.
| # of games | 70 | 50 | 25 | \(\cdots\) |
|---|---|---|---|---|
| # of games | 4.25 | 3.91 | 3.22 | \(\cdots\) |
However, results of an analysis might be difficult to interpret because the log of a measured variable is usually meaningless.
What other variables would you expect to be extremely skewed?
Salary, housing prices, ability to throw a football, …
What patterns are apparent in the change in population between 2000 and 2010?
A table that summarizes data for two categorical variables is called a contingency table.
The contingency table below shows the distribution of students' genders and whether or not they are looking for a spouse while in college.
| No | Yes | Total | |
|---|---|---|---|
| gender | |||
| Female | 86 | 51 | 137 |
| Male | 52 | 18 | 70 |
| Total | |||
| 138 | 69 | 207 | |
A bar plot is a common way to display a single categorical variable. A bar plot where proportions instead of frequencies are shown is called a relative frequency bar plot.
How are bar plots different than histograms? Bar plots are used for displaying distributions of categorical variables, while histograms are used for numerical variables. The x-axis in a histogram is a number line, hence the order of the bars cannot be changed, while in a bar plot the categories can be listed in any order (though some orderings make more sense than others, especially for ordinal variables.)
Does there appear to be a relationship between gender and whether the student is looking for a spouse in college?
| No | Yes | Total | |
|---|---|---|---|
| gender | |||
| Female | 86 | 51 | 137 |
| Male | 52 | 18 | 70 |
| Total | |||
| 138 | 69 | 207 | |
To answer this question we examine the row proportions:
What are the differences between the three visualizations shown below?
Can you tell which order encompasses the lowest percentage of mammal species?
Does there appear to be a relationship between class year and the number of clubs students are in?